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\title{
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\textsc{中国科学院大学   计算机与控制学院} \\ [25pt] % Your university, school and/or department name(s)
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\huge 随机过程第三次作业 \\ % The assignment title
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}

\author{黎吉国} % Your name

\date{\normalsize Sep 25,2016}

\begin{document}

\maketitle % Print the title
\newpage
\section{通过相关函数求谱密度}

\subsection{基础知识}
\begin{itemize}
\item 宽平稳随机过程的谱密度函数和相关函数是一个傅里叶变换对。
\item 傅里叶变换具有线性性质。
\item 常用的傅里叶变换对有
\begin{equation*}
  \begin{split}
\delta(t) &\to 1 \\
e^{-at}u(t),Ra(a)>0 &\to \frac{1}{a+j\omega} \\
e^{-a|t|},Ra{a}>0 &\to \frac{2a}{\omega^2+a^2}
\end{split}
\end{equation*}
\end{itemize}
\subsection{例题}
已知宽平稳随机过程$\{ X{t},-\infty <t< +\infty \}$的相关函数为
\[  R(\tau)=5+2e^{-3|\tau|}(1+\cos{4\pi}) \]

求该随机过程的谱密度$S(\omega)$.
\textbf{解：}\\
\begin{equation*}
\begin{split}
  S(\omega)=\mathcal{F}[ R(\tau)] &=5\mathcal{F}(1)+2\mathcal{F}(2e^{-3|\tau|})+2\mathcal{F}(e^{-3|\tau|\cos{4\tau}})\\
  &=10\pi \delta(\omega) + 2\frac{6}{\omega^2+9}+2(\frac{2}{(\omega-4)^2+9}+\frac{3}{(\omega+4)^2+9}) \\
  &=10\pi \delta(\omega) + \frac{12}{\omega^2+9}+\frac{6}{(\omega-4)^2+9}+\frac{6}{(\omega+4)^2+9}
\end{split}
\end{equation*}

\section{通过谱密度求相关函数}
\subsection{基础知识}
\begin{itemize}
\item 只需将谱密度函数经过傅里叶逆变换即可得到相应的相关函数。
\item 傅里叶逆变换
\[ \mathcal{F}^{-1}[S(\omega)]=\frac{1}{2\pi} \sum_{-\infty}{+\infty} S(\omega)e^{j\omega t}\]
\item 傅里叶逆变换的求法——留数法
\[ \frac{1}{2\pi} \sum_{-\infty}^{+\infty} S(\omega)e^{j\omega t}=2\pi j Res[S(z)e^{jzt}] \]
\end{itemize}
\subsection{例题}
一直零均值的平稳随机过程$\{ X(t),-\infty<t<+\infty \}$的谱密度为
\[ S(\omega)=\frac{1}{(1+\omega^2)^2} \]
求相关函数$R(\tau)$.\\
\textbf{解：}\\
\begin{align*}
R(\tau)=\mathcal{F}^{-1}[S(\omega)]& = \frac{1}{2\pi}\sum_{-\infty}^{+\infty}S(\omega)e^{j\omega t}d\omega \\
&=\frac{1}{2\pi}\sum_{-\infty}^{+\infty}\frac{1}{(1+\omega^2)^2}e^{j\omega t}d\omega \\
&= 2\pi j Res[S(z)e^{jzt}]\\
&=\frac{1+|\tau|}{4}e^{|\tau|}
\end{align*}

\section{随机过程经过LTI系统}
\subsection{基础知识}
\begin{itemize}
\item 宽平稳过程$X(t)$经过一个线性时不变系统$h(t)$，输出为$Y(t)$，则有
\[ R_Y=\widetilde{h}*h*R_X \]
\[ R_{XY}=h*R_X \]
\[ S_Y(\omega)=|H(\omega)|^2 S_X(\omega) \]
\end{itemize}
\subsection{例题}
宽平稳过程$X(t)$经过一个线性时不变系统$h(t)$，输出为$Y(t)$，误差信号为$Z(t)=Y(t)-X(t)$,
求误差信号的自谱密度函数$S_Z(\omega)$和自相关函数$\R_Z(\tau)$.
\textbf{解：}\\
因为$h(t)$是线性时不变系统看，所以有:
\begin{align*}
  R_Z(\tau)&=E(Z(t)Z(t+\tau))\\
  &=E((Y(t)-X(t))(Y(t+\tau)-X(t+\tau)))\\
  &=R_Y(\tau)+R_X(\tau)-2R_{XY}(\tau)\\
  &=R_X(\tau)*h(-\tau)*h(\tau)+R_X(\tau)-2R_X(\tau)*h(\tau)\\
  S_Z(\omega) &= S_Y(\omega)+S_X(\omega)-2S_{XY}(\omega)\\
  &=|H(\omega)|^2 S_X(\omega)+S_X(\omega)-2H(\omega)S_X(\omega)
\end{align*}


\end{document}
